SSBGVX(3)      LAPACK routine of NEC Numeric Library Collection      SSBGVX(3)



NAME
       SSBGVX

SYNOPSIS
       SUBROUTINE SSBGVX (JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q,
           LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
           INFO)



PURPOSE
            SSBGVX computes selected eigenvalues, and optionally, eigenvectors
            of a real generalized symmetric-definite banded eigenproblem, of
            the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
            and banded, and B is also positive definite.  Eigenvalues and
            eigenvectors can be selected by specifying either all eigenvalues,
            a range of values or a range of indices for the desired eigenvalues.




ARGUMENTS
           JOBZ      (input)
                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           RANGE     (input)
                     RANGE is CHARACTER*1
                     = 'A': all eigenvalues will be found.
                     = 'V': all eigenvalues in the half-open interval (VL,VU]
                            will be found.
                     = 'I': the IL-th through IU-th eigenvalues will be found.

           UPLO      (input)
                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N         (input)
                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           KA        (input)
                     KA is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

           KB        (input)
                     KB is INTEGER
                     The number of superdiagonals of the matrix B if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

           AB        (input/output)
                     AB is REAL array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first ka+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

                     On exit, the contents of AB are destroyed.

           LDAB      (input)
                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KA+1.

           BB        (input/output)
                     BB is REAL array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix B, stored in the first kb+1 rows of the array.  The
                     j-th column of B is stored in the j-th column of the array BB
                     as follows:
                     if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
                     if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

                     On exit, the factor S from the split Cholesky factorization
                     B = S**T*S, as returned by SPBSTF.

           LDBB      (input)
                     LDBB is INTEGER
                     The leading dimension of the array BB.  LDBB >= KB+1.

           Q         (output)
                     Q is REAL array, dimension (LDQ, N)
                     If JOBZ = 'V', the n-by-n matrix used in the reduction of
                     A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
                     and consequently C to tridiagonal form.
                     If JOBZ = 'N', the array Q is not referenced.

           LDQ       (input)
                     LDQ is INTEGER
                     The leading dimension of the array Q.  If JOBZ = 'N',
                     LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

           VL        (input)
                     VL is REAL

           VU        (input)
                     VU is REAL

                     If RANGE='V', the lower and upper bounds of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           IL        (input)
                     IL is INTEGER

           IU        (input)
                     IU is INTEGER

                     If RANGE='I', the indices (in ascending order) of the
                     smallest and largest eigenvalues to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           ABSTOL    (input)
                     ABSTOL is REAL
                     The absolute error tolerance for the eigenvalues.
                     An approximate eigenvalue is accepted as converged
                     when it is determined to lie in an interval [a,b]
                     of width less than or equal to

                             ABSTOL + EPS *   max( |a|,|b| ) ,

                     where EPS is the machine precision.  If ABSTOL is less than
                     or equal to zero, then  EPS*|T|  will be used in its place,
                     where |T| is the 1-norm of the tridiagonal matrix obtained
                     by reducing A to tridiagonal form.

                     Eigenvalues will be computed most accurately when ABSTOL is
                     set to twice the underflow threshold 2*SLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*SLAMCH('S').

           M         (output)
                     M is INTEGER
                     The total number of eigenvalues found.  0 <= M <= N.
                     If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

           W         (output)
                     W is REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z         (output)
                     Z is REAL array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                     eigenvectors, with the i-th column of Z holding the
                     eigenvector associated with W(i).  The eigenvectors are
                     normalized so Z**T*B*Z = I.
                     If JOBZ = 'N', then Z is not referenced.

           LDZ       (input)
                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK      (output)
                     WORK is REAL array, dimension (7N)

           IWORK     (output)
                     IWORK is INTEGER array, dimension (5N)

           IFAIL     (output)
                     IFAIL is INTEGER array, dimension (M)
                     If JOBZ = 'V', then if INFO = 0, the first M elements of
                     IFAIL are zero.  If INFO > 0, then IFAIL contains the
                     indices of the eigenvalues that failed to converge.
                     If JOBZ = 'N', then IFAIL is not referenced.

           INFO      (output)
                     INFO is INTEGER
                     = 0 : successful exit
                     < 0 : if INFO = -i, the i-th argument had an illegal value
                     <= N: if INFO = i, then i eigenvectors failed to converge.
                             Their indices are stored in IFAIL.
                     > N : SPBSTF returned an error code; i.e.,
                           if INFO = N + i, for 1 <= i <= N, then the leading
                           minor of order i of B is not positive definite.
                           The factorization of B could not be completed and
                           no eigenvalues or eigenvectors were computed.



LAPACK routine                  31 October 2017                      SSBGVX(3)